Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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### Fundamental representations and weight space dimension

For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all ...

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108 views

### Chern-Simons theory with non-compact gauge groups G

This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...

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117 views

### Littlewood Richardson Rule for general linear group over finite field

I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive ...

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### Defining a notion of “volume of its lattice” for non-rational subspaces

Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...

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103 views

### Nonlinear sigma models with non-compact groups / target spaces

A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...

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133 views

### When is G->G/H a trivial bundle

Suppose that $H\subseteq G$ are connected lie groups. Then $G\mapsto G/H$ is a principal $H$-bundle. I would like to know if there is some non-tautological characterization of when this is a trivial ...

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113 views

### Master thesis about Myers-Steenrod theorem

I hope here is the best place to ask this, anyway this year I have to write a master thesis in pure mathematics and since I'm interested in geometry and topology normally I will look for a topic about ...

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429 views

### Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...

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67 views

### Classification of Euclidian-like Klein geometries in spirit of Erlangen program

All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but ...

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89 views

### Differential operators on $G/K$

Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...

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141 views

### What kind of locally symmetric space is a rational sphere

Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...

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211 views

### Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)

It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...

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70 views

### A lie group which is sat in its Lie algebra

Motivated by this question we ask the follwing question:
Assume that a Lie subgroup $G$ of $Gl(n,\mathbb{R})$ is contained in a subvector space $F$ of $M_n(\mathbb{R})$ such that $dim G=...

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376 views

### Probability of satisfying a word in a compact group

This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. ...

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272 views

### Maps from 2-Torus to SO(3)

Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]